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The Analysis of Dispersion
for Trajectories of Fire-extinguishing Rocket
CRISTINA MIHAILESCU
Electromecanica –Ploiesti SA
Soseaua Ploiesti-Tirgoviste, Km 8
ROMANIA
[email protected] http://www.elmec.ro
MARIUS RADULESCU
Electromecanica –Ploiesti SA
Soseaua Ploiesti-Tirgoviste, Km 8
ROMANIA
[email protected] http://www.elmec.ro
FLORENTINA COMAN
Electromecanica –Ploiesti SA
Soseaua Ploiesti-Tirgoviste, Km 8
ROMANIA
[email protected] http://www.elmec.ro
Abstract: In this study a dispersion analysis has been completed on flight simulation software of an unguided
fire-extinguishing rocket in order to predict the trajectory parameters and the trajectory dispersion, to analyze
the probability of finding the impact point and also to determine the safety area which is a critical condition in
firing fields’ tests. The total dispersion results mainly due to manufacturing inaccuracy, measurement errors
which includes propellant mass, rocket total mass, axial and lateral moments of inertia and position of mass
centre, thrust and fin misalignments, errors due to aerodynamic coefficients prediction and atmospheric
modelling errors which include atmospheric disturbances such as fluctuations in wind profile, errors of
atmospheric properties like pressure, density, temperature.
Key-Words: Dispersion Analysis, Nominal Trajectory, Flight simulation, Fire-extinguishing rocket.
1 Introduction Dispersion is a measure for the deviation of the
rocket’s trajectory from the trajectory of some
standard rocket, called nominal trajectory. For a
rocket, dispersion arises from three different
sources: events that occur at launching, events
during burning after launching, and events after
burning. For rockets, most of the dispersion arises
during the burning period after launching. This is
predicted theoretically and is confirmed by
experience [1].
Simulation of the trajectory of unguided rockets has
as input data characteristics of the rocket,
atmospheric conditions and launching conditions.
In reality all these input data may have some
variations around a nominal value that implies a
variation in trajectory path. Therefore, there is
always an area of possible impact points. The only
way to minimize this impact area is to impose some
restrictions in manufacturing tolerances, since no
restrictions can be made in minimizing the errors of
atmospheric conditions. The aim of this paper is to
determine the total dispersion of trajectory and to
estimate the impact point using the probability
analyze method.
The parameters that induce dispersion of rocket’s
trajectory are:
• The propellant mass and composition
inaccuracy;
• The rocket total mass, axial and lateral
moments of inertia and resultant centre of gravity
inaccuracies;
• Launcher deflection;
• The thrust force of the rocket engine: because
of the tolerance in rocket engine design, propellant
properties, and manufacturing;
• Thrust and fin misalignments: it is an important
source of dispersion in case of unguided rockets.
Recent Advances in Fluid Mechanics and Heat & Mass Transfer
ISBN: 978-1-61804-026-8 135
• Atmospheric disturbances such as wind profile,
tail wind, cross wind, and gusts, variation in
atmospheric density
• Rocket characteristics, such as aerodynamic
coefficients which are previously calculated or
measured in wind tunnels.
All these parameters differ from their nominal
value and will generate errors derived from
measurement, manufacturing or modelling. These
sources of error are all mutually independent. Thus,
the composite errors are just their combination.
There are some methods to estimate the dispersion
of trajectory for a rocket:
• Root Mean Square Method
• Monte Carlo method
• Method of covariance matrix
The Root Mean Square Method simulates the
rocket trajectory perturbing one parameter at time
and the results are compared with the nominal
results. The sum of squares deviations for all
parameters is square of total deviation.
Monte Carlo method of dispersion removes smaller
dispersion parameters. Each input parameter is
selected randomly in the defined ranges and used in
the simulation of trajectory.
The Method of covariance matrix: In probability
theory and statistics, covariance is a measure of
how much two variables change together. A
covariance matrix is a matrix whose element in the
i, j position is the covariance between the i th and j
th
elements of a random vector (that is, of a vector of
random variables). Each element of the vector is a
scalar random variable, either with a finite number
of observed empirical values or with a finite or
infinite number of potential values specified by a
theoretical joint probability distribution of all the
random variables.
The parameters uncertainties, based on
measurement statistics and observations are
presented in table 1.
Table 1
No Parameter Variation
1 Mass ±1 %
2 X position of mass centre ±6 %
3 Thrust ±1 %
4 Wind velocity – Ox component ±10 m/s
5 Wind velocity – Oy component ±10 m/s
6 Wind velocity – Oz component ±10 m/s
7 Initial velocity ±2 m/s
8 Moment of inertia ±2%
9 Moment of inertia ±2%
10 Launching angle ±1 deg
All the calculations presented in the paper were
performed considering the first 6 parameters
variation, but the mathematical model accepts
variation for a larger number of parameters that
will generate dispersion.
2. Model Description In order to predict the trajectory of an unguided
rocket, six degrees of freedom (6-DOF)
mathematical model is used.
As a first approximation for the deviation of
trajectory from the nominal value we can
investigate the individual influence of one
parameter at a time. We’ll find some preliminary
limits for the trajectory.
Next, we can use possible combinations of
parameters to investigate their influence on
trajectory. The idea is to ensure that most of the
worst possible cases are considered. This is a
motivation to investigate the influence of a
combination of parameters. The analysis has been
done in two ways:
1. For the first model any parameter will pass
through its interval of variation with its own
defined step; each combination of parameters will
generate a new trajectory. The method accuracy
strongly depends on step density for every
parameter. Also we can associate a probability
density value to each parameter individually
considering a normal distribution (Gaussian).
In probability theory, the normal (or Gaussian)
distribution is a continuous probability distribution
that is often used as a first approximation to
describe real-valued random variables that tend to
cluster around a single mean value. The graph of
the associated probability density function is
known as the Gaussian function or bell curve (fig.
1): ( )
2
2
2
22
1)( σ
µ
πσ
−−
=
x
exf (1)
where parameter µ is the mean (location of the peak)
and σ2 is the variance (the measure of the width of
the distribution). The probability for the variable to
fall within a particular region is given by the
integral of this variable’s density over the region.
The probability density function is maxim around
the nominal value of the considered parameter,
decreasing to zero near the limits of the interval of
variation. The probability density function’s
integral over the entire space is equal to one.
Recent Advances in Fluid Mechanics and Heat & Mass Transfer
ISBN: 978-1-61804-026-8 136
Fig. 1 The graph of Gaussian function
We will obtain a value for the probability density
function for every parameter that supports
variations. A sum of N independent random
variables, with densities U1, …, UN:
( ) )(...)(11 ... xffxf
NN UUUU ⋅⋅=++ (2)
2. The second model is based on Monte Carlo
Method and consists of generating random values
for the variable parameters in the proper interval.
Each possible combination of values will generate a
new trajectory. This method’s accuracy strongly
depend on number of iterations, as we can see
further in the next section, fig.12 and fig.13.
As in the previous case we can associate a
probability density function to each parameter that
will generate a probability density function for
every point of the trajectory and in particular for
the impact point.
The configuration of the rocket has four fins and it
is presented in fig. 2. The body of the rocket is
cylindrical, with an over-calibre of the front part. A
full presentation of this kind of rocket and its utility
can be found in [2].
Fig. 2 The Fire-extinguishing Rocket’s
configuration
3. Simulation results In this study, a dispersion trajectory calculation
using a six-degree-of-freedom (6-DOF) model was
developed and applied for fire-extinguishing rocket
[3]. All aerodynamic forces and moments
coefficients of the configuration are previously
calculated, and they are input data. The mass
properties are calculated considering the change of
the rocket mass during propellant burning till the
propellant burn-out (active part), then the rocket
will fly the rest of its trajectory as a projectile of
fixed mass (passive part). The 6-DOF model
assumed the rocket is ideal, where the axis of
symmetry of the exterior surface coincides with the
longitudinal principal axis of inertia, and the two
lateral principal moments of inertia are identical.
3.1. Nominal trajectory Under the circumstances that no atmospheric
disturbances were present, no manufacturing,
reading or measuring errors were made, the
nominal trajectory have been generated. In order to
investigate the trajectory parameters of the given
unguided rocket, three cases will be chosen
corresponding to firing angles: 300, 40
0, 50
0. The
3D representation of nominal trajectories for
different launching angles is presented in fig. 3.
These cases will give us the possibility to compare
the results corresponding to firing angle.
X [m]
0
1000
2000
3000
4000
5000Y [m
]-4
-2
0
2
4
Z[m]
0
500
1000
1500
2000
Y
X
Z
launching angle:
50 deg
40 deg
30 deg
Fig. 3 Nominal trajectories for different launching
angles
As we can observe in fig.3 there is a lateral
deviation for the nominal trajectories due to rocket
rotation.
Active
section
Propulsion
system
Stabilizer
block
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ISBN: 978-1-61804-026-8 137
3.2 The individual influence of parameters To investigate the individual influence on trajectory,
every parameter will be modified inside the proper
variation interval. Using the results it was possible
to plot the impact point distance error vs. different
parameters variation.
The influence of 1% mass variation on impact point
is presented in fig. 4. Errors due to mass centre
uncertainty are presented in fig. 5. Figure 6 shows
the influence of 1% variation thrust on impact point
position. Figure 7 reflects the corresponding
deviations of impact point due to wind presence.
X [m]
Y[m]
4430 4440 4450 4460 44702.5
2.55
2.6
2.65
2.7
2.75
2.8
nominal value
increased with 1%
decreased with 1%
Fig. 4 The influence of mass variation on impact
point
X [m]
Y[m]
4400 4410 4420 4430 4440 4450 4460 4470 44801
1.5
2
2.5
3
3.5
4
nominal value
increased with 6%
decreased with 6%
Fig. 5 The influence of mass centre variation on
impact point
X [m]
Y[m]
4430 4440 4450 4460 44702.5
2.55
2.6
2.65
2.7
2.75
2.8
nominal value
increased with 1%
decreased with 1%
Fig. 6 The effect of the thrust error on impact point
X [m]
Y[m]
4000 4200 4400 4600 4800-60
-40
-20
0
20
40
60
No wind
Vy wind = 10 m/s
Vy wind = -10 m/s
Vz wind = -10 m/s
Vz wind = 10 m/s
Vx wind = 10 m/s
Vx wind = -10 m/s
Fig. 7 The effect of the wind velocity on impact
point position
The rocket range increased in presence of a tail
wind as shown in Fig. 7. Also, if a cross wind is
presented the rocket drifts to right if the wind came
from right and vice verse due to the presence of tail
surfaces behind the rocket centre of gravity to make
the rocket fly opposite to wind direction.
Considering all the individual deviations we can
obtain an approximation of total deviation. In table
2 there are presented all deviation obtained in the
previous example.
Table 2
Deviation No Parameter
X [m] Y [m]
1 Mass 22 0.07
2 X position of mass centre 42 1.6
3 Thrust 29 0.11
4 Ox wind velocity 520 22
5 Oy wind velocity 40 82
6 Oz wind velocity 330 18
Recent Advances in Fluid Mechanics and Heat & Mass Transfer
ISBN: 978-1-61804-026-8 138
The value of total deviation corresponding to the
case of 50 deg launching angle is around 630 m.
3.3 The combinations of parameters In the previous section it was presented an
individual analysis of variation for each parameter
at once. But combinations of parameters could
strongly influence the trajectory path. The results
have been obtained using the two models
previously described.
X [m]
Y[m]
3800 4000 4200 4400 4600 4800 5000 5200-600
-400
-200
0
200
400
6008.00E-01
7.00E-01
6.00E-01
5.00E-01
4.00E-01
3.00E-01
2.00E-01
1.00E-01
probability
Fig. 8 The area of possible impact point using the
first model
The second model used Monte Carlo Method. As
we have been expected, the impact probability is
higher near the nominal impact point position (fig.
8, 9, 10).
X [m]
Y[m]
3800 4000 4200 4400 4600 4800 5000 5200-600
-400
-200
0
200
400
600
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
probability
Fig. 9 The area of possible impact point using the
Monte Carlo method for 1000 iterations
X [m]
Y[m]
3800 4000 4200 4400 4600 4800 5000 5200-600
-400
-200
0
200
400
600
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
probability
Fig. 10 The area of possible impact point using the
Monte Carlo method for 10000 iterations
All the previous graphic results correspond to 50
deg launching angle. Similarly results for 30 deg
and 40 deg firing angles have been obtained and
they are presented in fig. 11 and 12. The maximum
dimensions of dispersion area on impact are
presented in table 3.
X [m]
Y[m]
3800 4000 4200 4400 4600 4800 5000 5200-600
-400
-200
0
200
400
600
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
probability
Fig. 11 Dispersion area for 30deg launching angle
X [m]
Y[m]
4000 4200 4400 4600 4800 5000 5200 5400-600
-400
-200
0
200
400
600
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
probability
Fig. 12 Dispersion area for 40deg launching angle
Recent Advances in Fluid Mechanics and Heat & Mass Transfer
ISBN: 978-1-61804-026-8 139
Table 3
Firing angle
[deg]
|Xmax-Xmin|
[m]
|Ymax-Ymin|
[m]
30 930 750
40 1000 810
50 1030 960
From table 3 it is clear that dispersion increases
with firing angle because dispersion depends on
flight time.
4. Conclusion: While for some types of rockets is very important
to study the dispersion of trajectory at high
altitudes (like rockets used to seed the clouds), for
the fire-extinguishing rocket the dispersion in
impact area is interesting to analyze. The reason is
that spreading of extinguishing substances is taking
place in the moment of impact. On the other hand,
in highland area, for the case of fire localised on
slopes, the dispersion on a certain altitude must be
estimated. The way to do this is to intersect the
fascicle of trajectories with the inclined plane of
hill or mountain.
The influence on trajectory of possible
combinations of parameters was considered using
two different models. The disadvantage of the first
model is the expensive computing time, but this
model will always capture the extremes. The
second model has the advantage of flexibility and
rapidity, but it depends on random values and did
not exclude the possibility to omit an important
case in analysis.
A very important result of this analysis for
estimating with high accuracy the possible impact
area, in case of normal working rocket is the safety
operation of these means of fire extinguishing.
This paper presents the possible dispersion of the
fire-extinguishing rocket in case of normal working.
We don’t take under consideration eventually
accidents due to rocket engine malfunction,
breaking of some structural or aerodynamic
elements or due to a possible explosion during
flight.
A similar study can be made to investigate the
influence of aerodynamic coefficients uncertainties
on the rocket trajectory path and on the impact
point.
Because of the results that have been obtaining for
three different firing angles we can make a
comparison of dispersion for these cases. The
conclusion is that dispersion increases with firing
angle.
Trajectory analysis of an unguided rocket using
simulation software was undertaken to show the
importance of this type of analysis in order to know
all parameters acting on the rocket during its flight
which may be useful to avoid flight mistakes. The
dispersion analysis is used to show the importance
of identifying the design weaknesses in margins of
specific parameters. Also, it is used to find out the
optimum values of the rocket parameters for lowest
impact point error.
The practical aims of this analysis are:
• Programming of firing tests;
• Finding of rocket’s impact point in field to
collect the evidences of rocket’s efficiency.
Results were presented for the selected conditions
in the form of dispersion. This analysis showed that
the parameters errors have a great effect on the
rocket range and its impact point error.
References:
[1] Jeffrey A. Isaacson, David R. Vaughan,
“Estimation and Prediction of Ballistic Missile
Trajectories”, RAND, 1996.
[2] Chelaru, T.-V., Mihailescu, C., Neagu, I.,
Radulescu, M. “The stability of operating
parameters for fire-extinguishing rocket motor”,
Proceedings of the International Conference on
Energy and Environment Technologies and
Equipment, EEETE '10, pp. 190-195.
[3] Chelaru, T.-V., Barbu, C. “Mathematical
model and technical solutions for multi-stage
sounding rockets, using the rotation angles”,
Proceedings of the 3rd International
Conference on Applied Mathematics,
Simulation, Modelling, ASM'09, Proceedings of
the 3rd International Conference on Circuits,
Systems and Signals, CSS'09, pp. 94-99.
[4] Bernard Etkin, “Dynamics of Atmospheric
Flight”, United States of America, John Wiley
& Sons, 1972.
[5] Jankovic, S., Gallant, J., Celens, E., "Dispersion
of an Artillery Projectile due to Unbalance",
18th International Symposium on Ballistics,
San Antonio, pp. 128-141, 15-19 Nov, 1999.
[6] Saghafi F., and Khalilidelshad M., "A Monte-
Carlo Dispersion Analysis of a Rocket Flight
Simulation Software", 17th European
Simulation Multi-Conference ESM 2003,
England, 9-11 June.
[7] Sailaranta, T., Siltavuori, A., Laine, S.,
Fagerstrom, B., "On Projectile Stability and
Firing Accuracy", 20th International Symposium
on Ballistics, Orlando, pp. 195-202, 23-27
September, 2002.
Recent Advances in Fluid Mechanics and Heat & Mass Transfer
ISBN: 978-1-61804-026-8 140
